Circulant preconditioners for pricing options. (English) Zbl 1233.91317
In this article, the authors investigate a non-symmetric Toeplitz system that arises when discretizing and solving the pricing PIDE of European call options in Merton’s jump-diffusion model. If the jump distribution in Merton’s jump-diffusion model has mean \(\mu_J = 0\), then the Toeplitz system is symmetric and has been investigated in other papers. The authors look at the more general case \(\mu_J\neq 0\) and solve the resulting non-symmetric system using Strang’ circulant preconditioner in th enormalized pre-conditioned conjugate gradient method. The main result is that the spectrum is clustered around one and the smallest eigenvalue is uniformly bounded away from zero, leading to superlinear convergence of the conjugate gradient method. This is also highlighted using some numerical examples.
Reviewer: Reinhold Kainhofer (Wien)
MSC:
| 91G60 | Numerical methods (including Monte Carlo methods) |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 65F10 | Iterative numerical methods for linear systems |
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
Strang’s circulant preconditioner; nonsymmetric Toeplitz system; European call option; partial integro-differential equation; normalized preconditioned system; family of generating functionsReferences:
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